Proving AB and BA are Square Matrices

In summary, if the matrix products AB and BA are both defined, then AB and BA are square matrices. Additionally, if A is an m*n matrix and A(BA) is defined, then B is an n*m matrix. These statements serve as proofs for the relationship between matrix products and the dimensions of their matrices.
  • #1
forty
135
0
1)
Show that if matrix products AB and BA are both defines, then AB and BA are square matrices:



Let A = a m*n matrix

IF AB is defined then B must have n rows (n*?) matrix

IF BA is defined then B must have m columns making it a n*m matrix

so BA = (n*m) * (m*n) = (n*n) matrix

AB = (m*n) * (n*m) = (m*m) matrix



2)
Show that if A is an m*n matrix and A(BA) is defined, then B is an n*m matrix.



IF BA is defined and A is m*n matrix then B must be a ?*m matrix

BA produces a ?*n matrix

IF A(BA) is defined BA must be a n*n matrix

AS BA is an n*n matrix B must be a n*m matrix


Do these work as a proofs, if they even follow any logic in the first place (I'm horrid when it comes to matrices)

Any input would be greatly appreciated :)
 
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  • #2
They look perfectly good to me.
 
  • #3
love you :)
 

1. How do you prove that AB and BA are square matrices?

To prove that AB and BA are square matrices, you must first show that both matrices have the same number of rows and columns. This means that they must be of the form nxn, where n is the number of rows and columns. Then, you must show that the product of these matrices is also a square matrix, which means that the resulting matrix must also be of the form nxn.

2. What is the significance of proving that AB and BA are square matrices?

Proving that AB and BA are square matrices is important because it demonstrates the commutative property of matrix multiplication. This means that the order of multiplying two square matrices does not affect the result, which is not always the case with other types of matrices.

3. Can you provide an example of how to prove that AB and BA are square matrices?

Yes, for example, if we have two 2x2 matrices: A = [1 2; 3 4] and B = [5 6; 7 8], we can show that both AB and BA are square matrices by calculating the product of the matrices. AB = [19 22; 43 50] and BA = [23 34; 31 46], which are both 2x2 matrices.

4. What is the difference between proving AB and BA are square matrices and proving that AB = BA?

Proving that AB and BA are square matrices simply shows that the resulting matrix is of the same size, while proving that AB = BA means that the actual values in the matrices are also equal. In other words, proving AB = BA is a stronger statement, as it shows that the matrices are not only the same size, but also have the same values in each corresponding position.

5. Are there any exceptions to the commutative property of matrix multiplication for square matrices?

Yes, there are exceptions to the commutative property of matrix multiplication for square matrices. This property only holds true if the matrices are invertible, meaning that they have an inverse matrix. If the matrices are not invertible, then the order of multiplication can affect the result.

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